Q1. Solve the system of equations using substitution:

Background

Topic: Solving Systems of Linear Equations (Substitution Method)

This question tests your ability to solve a system of two linear equations by substitution, a fundamental skill in algebra.

Key Terms and Formulas:

• System of equations: Two or more equations with the same variables.

• Substitution method: Solve one equation for one variable, then substitute that expression into the other equation.

Step-by-Step Guidance

1. Solve the first equation for (already done): .

2. Set the two expressions for equal to each other: .

3. To eliminate fractions, multiply both sides by 4.

4. Combine like terms to isolate on one side of the equation.

5. Once you have , substitute it back into one of the original equations to solve for .

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Q2. Solve the system by any method: graphing, substitution, or elimination.

Background

Topic: Solving Systems of Linear Equations

This question tests your ability to solve a system of equations using your preferred method: graphing, substitution, or elimination.

Key Terms and Formulas:

• Elimination method: Add or subtract equations to eliminate one variable.

• Substitution method: Solve one equation for one variable and substitute into the other.

Step-by-Step Guidance

1. Consider multiplying the second equation by 5 to eliminate the fraction.

2. Rewrite the second equation: and $8x - 5y = 40$ (after multiplying).

3. Compare the two equations to see if they are equivalent or contradictory.

4. Analyze whether the system has one solution, no solution, or infinitely many solutions based on the comparison.

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Q3. A biker bikes into the wind for 60 miles in 6 hours, then returns with the wind in 5 hours. Find the speed in still air and the wind speed.

Background

Topic: Systems of Equations (Distance, Rate, and Time)

This problem involves setting up and solving a system of equations based on the relationship , considering the effect of wind on speed.

Key Terms and Formulas:

• Let = speed in still air, = wind speed.

• With wind: ; Against wind: .

• Distance = Rate × Time:

Step-by-Step Guidance

1. Set up two equations:

   • Into the wind:

   • With the wind:

2. Solve each equation for and .

3. Add the two equations to eliminate and solve for .

4. Substitute back into one equation to solve for .

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Q4. A nursery mixes rye seed ($4.75/lb) and bluegrass seed ($2.50/lb) to make 120 lbs of a blend selling for $3.81/lb. How much of each type was used?

Background

Topic: Mixture Problems (Systems of Equations)

This question tests your ability to set up and solve a system of equations for mixture problems involving cost and quantity.

Key Terms and Formulas:

• Let = pounds of rye seed, = pounds of bluegrass seed.

• Total weight:

• Total cost:

Step-by-Step Guidance

1. Write the two equations based on the information given.

2. Solve the first equation for one variable (e.g., ).

3. Substitute this expression into the second equation.

4. Solve for , then use the result to find .

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Q5. Graph the system of linear inequalities: ,

Background

Topic: Graphing Systems of Linear Inequalities

This question tests your ability to graph solution regions for systems of inequalities.

Key Terms and Formulas:

• Linear inequality: An inequality involving a linear function.

• Solution region: The set of points that satisfy all inequalities in the system.

Step-by-Step Guidance

1. Graph the boundary line for (the y-axis). Since the inequality is , shade to the right of the y-axis.

2. Graph the line . Since the inequality is , shade below or on the line.

3. The solution region is where the shaded areas overlap.

4. Remember to use a dashed line for if the inequality is strict (), and a solid line for since the inequality is .

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Q6. Graph the system of linear inequalities: ,

Background

Topic: Graphing Systems of Linear Inequalities

This question tests your ability to graph and find the solution region for two inequalities.

Key Terms and Formulas:

• Graph each boundary line: and .

• Shade the appropriate side for each inequality.

Step-by-Step Guidance

1. Rewrite each inequality in slope-intercept form () to make graphing easier.

2. Graph the boundary lines, using a dashed line for and a solid line for .

3. Shade the region above the line for and below the line for .

4. The solution region is where the shaded areas overlap.

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Q7. Determine if the expression is a polynomial. If so, write in standard form, state the degree, and classify it.

Background

Topic: Polynomials (Classification and Standard Form)

This question tests your understanding of what constitutes a polynomial, how to write it in standard form, and how to determine its degree and classification.

Key Terms and Formulas:

• Polynomial: An expression consisting of variables and coefficients, using only addition, subtraction, and multiplication, and non-negative integer exponents.

• Standard form: Terms are written in descending order of degree.

• Degree: The highest sum of exponents in any term.

• Classification: Monomial (1 term), Binomial (2 terms), Trinomial (3 terms).

Step-by-Step Guidance

1. Check if all exponents are non-negative integers and variables are combined by addition, subtraction, or multiplication.

2. Write the terms in descending order of degree.

3. Find the degree by adding exponents in each term and identifying the largest sum.

4. Count the number of terms to classify as monomial, binomial, or trinomial.

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